1. The problem asks us to multiply the expressions $2^{1/5}$ and $2^{2/5}$ and express the result in radical form. 2. Recall the exponent rule: when multiplying powers with the same base, add the exponents. 3. Calculate the sum of the exponents: $$\frac{1}{5} + \frac{2}{5} = \frac{3}{5}$$ 4. Rewrite the product using the sum of exponents: $$2^{1/5} \times 2^{2/5} = 2^{3/5}$$ 5. Express $2^{3/5}$ in radical form: $$2^{3/5} = \sqrt[5]{2^3} = \sqrt[5]{8}$$ 6. So, the product in radical form is $\sqrt[5]{8}$.